Process capability indices for one‐sided specification intervals and skewed distributions

Abstract

One-sided specification intervals are frequent in industry, but the process capability analysis is not well developed theoretically for this case. Most of the published articles about process capability focus on the case when the specification interval is two-sided. Furthermore, usually the assumption of normality is necessary. However, a common practical situation is process capability analysis when the studied characteristic has a skewed distribution with a long tail towards large values and an upper specification limit only exists. In such situations it is not uncommon that the smallest possible value of the characteristic is 0 and that this also is the best value to obtain. We propose a new class of indices for such a situation with an upper specification limit, a target value zero, and where the studied characteristic has a skewed, zero-bound distribution with a long tail towards large values. A confidence interval for an index in the proposed class, as well as a decision procedure for deeming a process as capable or not, is discussed. These results are based on large sample properties of the distribution of a suggested estimator of the index. A simulation study is performed, assuming the quality characteristic is Weibull distributed, to investigate the properties of the suggested decision procedure.

Skewed zero-bound distributions and process capability indices for upper specifications

“…As a consequence, the index C M Np (v) in Equation (3) is not suitable for our studied situation and needs to be modified. Using the same reasoning as in Vännman and Albing [27] led to the same class of indices C MA (τ, v), for v > 0, where…”

“…The larger the index value, the more capable the process in the sense that either the quantile q 1−τ has decreased or the median has moved closer to 0. The class of indices C MA (τ, v) in Equation (4) possesses a number of desirable properties, as discussed by Vännman and Albing [27]. For example, for a given value greater than or equal to 1 the indices limit the probability of non-conformance to be less than or equal to τ .…”

“…It can be noted that in such situations, when trying to transform the data to normally distributed data using a suitable Box-Cox transformation with power λ ≤ 0 for positively skewed distributions (see [2]), the target value goes to infinity. So far, the only published index found for the studied situation is presented by Vännman and Albing [27]. Here we try a new approach to find a suitable index and arrive at the same class of indices as proposed by Vännman and Albing [27].…”

“…So far, the only published index found for the studied situation is presented by Vännman and Albing [27]. Here we try a new approach to find a suitable index and arrive at the same class of indices as proposed by Vännman and Albing [27]. For this class of indices, we consider two different estimators and derive the asymptotic distributions for these two estimators.…”

“…Vännman and Albing [27] applied the quantile approach by Clements'starting from another family of indices, introduced by Vännman [25] and designed for one-sided specification intervals with Downloaded by [Northeastern University] at 12:24 04 November 2014 target values but based on the normality assumption. They then arrived at a class of indices, C * pνu (u, v), which includes Equation (3) as a special case.…”